3.27.83 \(\int \frac {\sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\) [2683]

Optimal. Leaf size=158 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {13}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {36}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1409 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {1091 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125 \sqrt {33}} \]

[Out]

-1409/9375*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1091/103125*EllipticF(1/7*21^(1/2)*(
1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5*(2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)+36/125*(2+3*x)^(3/2)*(1-2
*x)^(1/2)*(3+5*x)^(1/2)+13/625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \begin {gather*} -\frac {1091 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125 \sqrt {33}}-\frac {1409 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {2 \sqrt {1-2 x} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}+\frac {36}{125} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {13}{625} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(3 + 5*x)^(3/2),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (13*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/625 + (3
6*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/125 - (1409*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/3125 - (1091*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3125*Sqrt[33])

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (\frac {11}{2}-18 x\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {36}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {2}{125} \int \frac {\sqrt {2+3 x} \left (-50+\frac {39 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {13}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {36}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2 \int \frac {\frac {5727}{4}+\frac {4227 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1875}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {13}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {36}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1091 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6250}+\frac {1409 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3125}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {13}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {36}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1409 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {1091 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 4.02, size = 102, normalized size = 0.65 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (119+485 x+450 x^2\right )}{\sqrt {3+5 x}}+2818 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+455 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{18750} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(3 + 5*x)^(3/2),x]

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(119 + 485*x + 450*x^2))/Sqrt[3 + 5*x] + 2818*Sqrt[2]*EllipticE[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2] + 455*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/18750

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Maple [A]
time = 0.09, size = 143, normalized size = 0.91

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (3273 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2818 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-81000 x^{4}-100800 x^{3}-8970 x^{2}+25530 x +7140\right )}{18750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(143\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {18 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}+\frac {43 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {1909 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1409 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{13125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/18750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3273*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*El
lipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-2818*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/
7*(28+42*x)^(1/2),1/2*70^(1/2))-81000*x^4-100800*x^3-8970*x^2+25530*x+7140)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(5/2)*sqrt(-2*x + 1)/(5*x + 3)^(3/2), x)

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Fricas [A]
time = 0.12, size = 33, normalized size = 0.21 \begin {gather*} \frac {{\left (450 \, x^{2} + 485 \, x + 119\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{625 \, \sqrt {5 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/625*(450*x^2 + 485*x + 119)*sqrt(3*x + 2)*sqrt(-2*x + 1)/sqrt(5*x + 3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**(5/2)*(1-2*x)**(1/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(5/2)*sqrt(-2*x + 1)/(5*x + 3)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(3*x + 2)^(5/2))/(5*x + 3)^(3/2),x)

[Out]

int(((1 - 2*x)^(1/2)*(3*x + 2)^(5/2))/(5*x + 3)^(3/2), x)

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